Firm Dynamics, Endogenous Markups and the Labor Share of

Income�

Andrea Colciago

De Nederlandsche Bank

and University of Milano Bicocca

Lorenza Rossi

University of Pavia

Abstract

Recent U.S. evidence suggests that the response of labor share to a productivity shock

is characterized by countercyclicality and overshooting. These �ndings cannot be easily rec-

onciled with existing business cycle models. We extend the Diamond-Mortensen-Pissarides

model of search in the labor market by considering strategic interactions among an en-

dogenous number of producers, which leads to countercyclical price markups. While Nash

bargaining delivers a countercyclical labor share, we show that countercyclical markups

are fundamental to address the overshooting. On the contrary, we �nd that real wage

rigidity does not seem to play a crucial role for the dynamics of the labor share of income.

JEL Classi�cation Numbers: E24, E32, L11.

Keywords: Labor Share Overshooting, Endogenous Market Structures, Search and Match-

ing Frictions.

�We are grateful to seminar participants at the Dutch National Bank, the Central Bank of Finland, the

University of Milano Bicocca, the Catholic University of Milano and the Kiel Institute for the World Economy.

Anton Cheremukhin, Diego Comin, Martin Ellison, Federico Etro, Stefano Gnocchi, Bill Kerr, Anton Nakov,

Tiziano Ropele, Patrizio Tirelli, Aleh Tsivinsky, Juuso Vanhala, Neeltje Van Horen and Jouko Vilmunen

provided insightful discussions on this topic. Lorenza Rossi thanks the Foundation Alma Mater Ticinensis for

�nancial support through the research grant "Promuovere la ricerca d�eccellenza". Correspondence: Lorenza

Rossi, University of Pavia, Department of Economics and Business, via San Felice 5, Pavia 27100, Italy.

1 Introduction

Figure 1 shows the dynamics of the labor share, the average product of labor and the real

wage to a one standard deviation orthogonalized productivity innovation for the U.S. in the

period 1954.I�2004.IV. Each response function is obtained from a bivariate VAR of order 1,

between the variable of interest and the Solow residual. The identi�cation assumption is that

the variable of interest has no contemporaneous e¤ect of the Solow residual.

As argued by Rios-Rull and Santaeulàlia-Llopis (2010), the response of the labor share

is characterized by countercyclicality and overshooting. The labor share falls on impact in

response to the shock and then shows an hump-shaped response, overshooting its long-run

level after �ve quarters, and peaking at the �fth year at a level larger in absolute terms than

the initial drop. Seven years after the peak the labor share is still half-way toward its steady

state value.

A model should satisfy two desiderata in order to account for the response of the labor

share to a technology shock displayed in the �gure. The �rst one is that the impact increase

in the real wage must be lower than that of average labor productivity. The second one is the

presence of a persistent wedge between average labor productivity and the real wage, such that

the response of the latter raises above that of the former for several periods. The �rst property

implies a countercyclical labor share, while the second one is necessary for overshooting.

Figure 1: Empirical IRFs of wages, average product of labor, and labor share to productivity

innovations in the U.S. Percentage deviations from long run averages. Source: Rios-Rull and

Santaeulàlia-Llopis (2010).

In this paper we build on Colciago and Rossi (2011) to develop a theory of the joint dy-

namics of the labor share and technology shocks which satis�es both desiderata and replicates

the countercyclicality and the overshooting of the labor share.

1

As argued by Rios-Rull and Santaeulàlia-Llopis (2010), standard business cycle models

cannot explain these empirical regularities. The RBC model implies that the real wage and

labor productivity move identically, so that the labor share of income displays no cyclical

dynamics. The conventional Diamond-Mortensen-Pissarides model (DMP model, henceforth)

of search in the labor market with Nash bargaining explains the countercyclicality of the labor

share in response to a productivity shock, but cannot address the overshooting.1 While the

overshooting of the labor share is still unexplained, targeting the dynamics of the labor share

in DSGE estimated models can help the identi�cation of relevant parameters.

We outline a DMP model with Nash Bargaining and Endogenous Market Structures. Mar-

ket structures are said to be endogenous since both the number of producers and price markups

are determined in each period. The model features �rms� entry à la Bilbiie, Ghironi and

Melitz (2012) (BGM 2012, henceforth) and oligopolistic competition between producers as

in Jaimovich and Floetotto (2008) and Colciago and Etro (2010). Nash bargaining allows

to replicate the countercyclicality of the labor share, while the key ingredient to replicate

the overshooting result is the countercyclicality of price markups originating from strategic

interactions between an endogenous number of producers. To build intuition, consider the

e¤ect of a technology shock. The latter creates pro�ts opportunities which attract �rms into

the market. This strengthens competition and, via strategic interactions, reduces persistently

the price markup. A persistently lower markup acts as a shifter of the standard marginal

product of labor and creates a wedge between average labor productivity and the real wage.

Speci�cally, a persistently lower price markup implies that the real wage rises relative to the

average productivity of labor for several periods. Besides being consistent with the dynamics

displayed in Figure 1, this leads to the overshooting of the labor share.

Aggregate real wages are characterized by an high degree of persistence. Hall (2005), inter

alia, points out that real wage rigidity is a feature needed to account for a number of labor

market facts. For this reason we study the e¤ect of real wage rigidity on the dynamics of the

labor share. Introducing real wage rigidity in the DMP framework with constant markups is

not su¢ cient to match the empirical evidence on the dynamics of the labor share in response

to a technology shock. We �nd that augmenting our framework with (a limited degree of)

real wage rigidity does not alter the previous �ndings, and allows a better matching of the

amplitude of the labor share overshooting observed in the data.

To the best of our knowledge we are the �rst to present a model addressing the over-

shooting of the labor share through countercyclical markups. Hornstein (1993) augments the

1Chois and Rios-Rull (2008), consider alternative search and matching models with Nash bargaining and

show that none of these models can replicate the labor share overshooting. Further, Rios - Rull and Santaeulàlia-

Llopis (2010), notice that the departure from a Cobb-Douglas technology is a necessary but not su¢ cient

condition to get the labor share overshooting.

2

neoclassical growth model with increasing return to scale, a �xed number of �rms and constant

markups. He �nds a labor share that is half as volatile as what is observed in the data, but

does not address the overshooting. Also, the role of real wage rigidities for the dynamics of

the labor share had not been explored yet.

Choi and Rios-Rull (2010) obtain the overshooting considering a model with putty-clay

technology, decentralized non-competitive wage setting (bilateral Nash bargaining) and an

aggregate technological shock that has a stronger e¤ect for newer hires. The technology process

that we adopt is, instead, fully standard. Shao and Silos (2011) also consider an economy

with costly entry of �rms and a frictional labor market. However, their model is characterized

by monopolistic competition between small �rms and by constant price markups. In their

framework the overshooting is due to the countercyclical value of vacancies. Nevertheless, this

condition is di¢ cult to test empirically. On the contrary, our transmission mechanism is well

supported by the empirical evidence. Bils (1987), Rotemberg and Woodford (2000) and Galì

et al. (2007) forcefully document price markup countercyclicality.

The remainder of the paper is organized as follows. Section 2 provides a decomposition

of the labor share of income. Section 3 outlines the model economy. Section 4 is devoted to

calibration. Section 5 contains the main results. Section 6 concludes. Technical details are

left in the Appendix.

2 The labor share and its components

Independently of the speci�cation of the model considered, the labor share is de�ned as lst =wtHtYt

= wtAt, where Ht are total hours worked and At = YtHtis the average productivity of

labor. In log-deviations blst = wt ��yt � Ht

�= wt � bAt; (1)

where a hat over a variable denotes the log-deviation from the steady state. Equation (1)

simply states that the log-deviation of the labor share is the di¤erence between the log-

deviation of the real wage and that of the average labor productivity. In the standard RBC

model the real wage equals the marginal product of labor. In log-deviation this amounts to

wt = yt � Ht = bAt (2)

As a result the labor share is constant and does not deviate from its steady state, that isblst = 0: Equations (1) and (2) suggest that in order to obtain a non constant labor share theallocative role of the real wage has to be broken.

In the search and matching framework this is obtained through Nash bargaining. The

latter implies that workers and �rms split the total surplus originating from a match. The

equilibrium real wage maximizes the joint surplus of the parties and depends on their relative

3

bargaining power. Thus, in the aftermath of a productivity increase just a fraction of the

latter is distributed to workers. Di¤erently from the standard RBC model, this implies that

the real wage rises by less than the increment in labor productivity. Hence, Nash Bargaining

helps explaining the countercyclicality of the labor share.

However, in the reminder we show that in the standard DMP framework with Nash bar-

gaining the real wage never raises relative to labor productivity in response to a technology

shock. This goes against the evidence reported in Figure 1 and, importantly, prevents the

standard DMP model from addressing the overshooting of the labor share.

In order to reproduce the overshooting, the real wage most rise relative to labor produc-

tivity for several periods. The countercyclical and inertial dynamics of price markup which

characterizes our approach delivers this mechanism.

3 The model

3.1 Labor and Goods Markets

There are two main building blocks in the model: oligopolistic competition with endogenous

entry in the goods market and search and matching frictions in the labor market. In this

paragraph we outlay their main features.

As in Colciago and Etro (2010) the economy features a continuum of sectors, or industries,

on the unit interval. Sectors are indexed with j 2 (0; 1) : Each sector j is characterized bydi¤erent �rms i = 1; 2; :::; Njt producing the same good in di¤erent varieties. At the beginning

of each period N ejt new �rms enter into sector j, while at the end of the period a fraction

� 2 (0; 1) of market participants exits from the market for exogenous reasons.

The labor market is characterized by search and matching frictions, as in Andolfatto

(1996) and Merz (1995). A fraction ut of the unit mass population is unemployed at time

t and searches for a job. Firms producing at time t need to post vacancies in order to hire

new workers. Unemployed workers and vacancies combine according to a CRS matching

function and deliver mt new hires, or matches, in each period. The matching function reads

as mt = m�vtott�1�

u t , where m re�ects the e¢ ciency of the matching process, vtott is the

total number of vacancies created at time t and ut is the unemployment rate. The probability

that a �rm �lls a vacancy is given by qt = mt

vtott, while the probability to �nd a job for an

unemployed worker reads as zt = mtut. Firms and individuals take both probabilities as given.

Matches become productive in the same period in which they are formed. Each �rm separates

exogenously from a fraction 1 � % of existing workers each period, where % is the probability

that a worker stays with a �rm until the next period.

As a result a worker may separate from a job for two reasons: either because the �rm

4

where the job is located exits from the market or because the match is destroyed. Since these

sources of separation are independent, the evolution of aggregate employment, Lt, is given by

Lt = (1� �) %Lt�1+mt: Thus, the number of unemployed workers searching for a job at time

t is ut = 1� Lt�1.

3.2 Households and Firms

Using the family construct of Mertz (1995) we can refer to a representative household consisting

of a continuum of individuals of mass one. Members of the household insure each other against

the risk of being unemployed. The representative family has lifetime utility:

U = E0

1Xt=0

�t

(Z 1

0lnCjtdj � �Lt

h1+1='t

1 + 1='

)�; ' � 0 (3)

where � 2 (0; 1) is the discount factor and the variable ht represents individual hours worked.Note that Cjt is a consumption index for a set of goods produced in sectors j 2 [0; 1], de�nedas

Cjt = N1

1�"jt

24NjtXi=1

Cjt(i)"�1"

35 ""�1

(4)

where Cjt(i) is the production of �rm i of this sector, and " > 1 is the elasticity of substitution

between the goods produced in each sector.2 The distinction between di¤erent sectors and

di¤erent goods within a sector allows to realistically separate limited substitutability at the

aggregated level, and high substitutability at the disaggregated level. The family receives

real labor income wthtLt and pro�ts from the ownership of �rms. Further, we assume that

unemployed individuals receive an unemployment bene�t b in real terms, leading to an overall

bene�t for the household equal to b (1� Lt). This is �nanced through lump sum taxation by

the government. Notice that the household recognizes that employment is determined by the

�ows of its members into and out of employment according to

Lt = (1� �) %Lt�1 + ztut (5)

Households choose how much to save in riskless bonds and in the creation of new �rms through

the stock market according to standard Euler and asset pricing equations.3

Each �rm i in sector j produces a good with a linear production function. We abstract

from capital accumulation issues and assume that labor is the only input. Output of �rm i in

2The term N1

1�"jt in (4) implies that there is no variety e¤ect in the model. However, allowing for a variety

e¤ect would not change our results.3These conditions are in the Appendix.

5

sector j is then:

yjt(i) = Atnjt (i)hjt(i) (6)

where At is the, common to all sectors, total factor productivity at time t, njt (i) is �rm i�s time

t workforce and hjt(i) represent hours per employee. Since each sector can be characterized in

the same way, in what follows we will drop the index j and refer to the representative sector.

3.3 Endogenous Market Structures

Following BGM (2012) we assume that new entrants at time t will only start producing at

time t + 1. Given the exogenous exit probability �, the average number of �rms per sector,

Nt, follows the equation of motion:

Nt+1 = (1� �)(Nt +N et ) (7)

where N et is the average number of new entrants at time t. In each period, the same nominal

expenditure for each sector EXPt is allocated across the available goods according to the

direct demand function:

yt(i) =

�pt(i)

Pt

��" YtNt

=pt(i)

�"

P 1�"t

EXPtNt

i = 1; 2; :::; Njt (8)

where Pt is the price index

Pt = N "�1jt

"NtXi=1

(pt (i))1�"# 11�"

(9)

such that total expenditure, EXPt, satis�es EXPt =NtXj=1

pt(j)yt(j) = PtYt.4 Inverting the

direct demand functions, we can derive the system of inverse demand functions

pt(i) =yt(i)

� 1"

NtXj=1

yt(j)"�1"

EXPt i = 1; 2; :::; Njt (10)

which will be useful for the derivation of the Cournot equilibrium. Period t real pro�ts of an

incumbent producer are de�ned as

�t (i) =pt (i)

Ptyt (i)� wt (i)nt (i)ht (i)� �vt (i) (11)

where wt (i) is the real wage paid by �rm i, vt (i) represents the number of vacancies posted at

time t and � is the output cost of keeping a vacancy open. The value of a �rm is the expected

4The demand of the individual good and the price index are the solution to the, usual, consumption expen-

diture minimization problem.

6

discounted value of its future pro�ts

Vt (i) = Et

1Xs=t+1

�t;s�s (i) (12)

where �t;t+1 = (1� �)��Ct+1Ct

��1is the households�stochastic discount factor which takes

into account that �rms�survival probability is 1� �. Incumbent �rms which do not exit fromthe market have a time t individual workforce given by

nt (i) = %nt�1 (i) + vt (i) qt (13)

Under di¤erent forms of competition between �rms we obtain prices satisfying:

pt (i)

Pt= �(";Nt)mct (i) (14)

where �(�;Nt) > 1 is the markup depending on the degree of substitutability between goods,

", and on the number of �rms, Nt, and mct (i) is the real marginal cost. In the remainder

of this section we characterize this mark up under Bertrand and Cournot competition taking

strategic interactions into account.

3.3.1 Bertrand Competition

Each �rm chooses pt (i) ; nt (i) and vt (i) to maximize �t (i) + Vt (i), taking as given the price

of the other �rms in the sector. The problem is subject to two constraints, namely equation

(8) and (13).5 The symmetric Bertrand equilibrium generates an equilibrium markup

�Pt (";Nt) =" (Nt � 1) + 1("� 1) (Nt � 1)

(15)

The markup �Pt is decreasing in the degree of substitutability between products ", with an

elasticity �P" = "Nt=(1 � " + "Nt)(" � 1). Moreover, the markup vanishes in case of perfectsubstitutability: lim"!1 �P (�;Nt) = 1. Finally, the markup is decreasing in the number of

�rms, with an elasticity �PN = N= [1 + "(N � 1)] (N � 1). Notice that the elasticity of themarkup to entry under competition in prices is decreasing in the level of substitutability

between goods, and it tends to zero when the goods are approximately homogenous. When

Nt ! 1 the markup tends to "=("� 1), the traditional one under monopolistic competition.As well known, strategic interactions between a �nite number of �rms lead to a higher markup

than under monopolistic competition.

5Details concerning the �rm maximization problem under Bertrand and Cournot competition are in the

Appendix.

7

3.3.2 Cournot Competition

In this case �rms maximize �t (i) + Vt (i) choosing their production yt(i) beside nt (i) and

vt (i) ; taking as given the production of the other �rms. The pro�t maximization problem

is constrained by the inverse demand function (10) and by equation (13). The symmetric

Cournot equilibrium generates a equilibrium markup

�Q(";Nt) ="Nt

("� 1) (Nt � 1): (16)

First of all notice that for a given number of �rms, the markup under competition in

quantities is always larger than the one obtained under competition in prices.6 Further, also

in this case the markup is decreasing in the degree of substitutability between products ", with

an elasticity �Q" = 1=("� 1), which is always smaller than �P" : higher substitutability reducesmarkups faster under competition in prices. In the Cournot equilibrium, the markup remains

positive for any degree of substitutability, since even in the case of homogenous goods, we

have lim"!1 �Q(";Nt) = Nt=(Nt � 1). The markup �Q(";Nt) is decreasing and convex inthe number of �rms with elasticity �QN = 1=(N � 1), which is decreasing in Nt (the markupdecreases with entry at an increasing rate) and independent from the degree of substitutability

between goods. Since �QN > �PN for any number of �rms or degree of substitutability, entry

decreases markups faster under competition in quantities compared to competition in prices,

a result that will impact on the relative behavior of the economy under the two forms of

competition. Only when Nt ! 1 the markup tends to "=(" � 1), which is the traditionalmarkup under monopolistic competition.

3.4 Entry and Job creation

We assume that entry requires a �xed cost , which is measured in units of output. De�ne

V et as the value at time t of a prospective entrant. Given our timing assumption, the latter

represents the value of a �rm which will start producing at time t+1. In each period the level

of entry is determined endogenously to equate the value of a prospective entrant to the entry

cost7

V et = (17)

Pro�ts maximization implies the following Job Creation Condition (JCC)

�

qt=

1

�jt� wtAt

!Atht + %Et�t;t+1

�

qt+1

6This is well known for models of product di¤erentiation (see for instance Vives, 1999).7This condition holds as long as the mass of new entrants Ne

t is positive. As Bilbiee, Ghironi and Melitz

(2012), we assume that macroeconomic shocks are small enough for this condition to hold in each period.

8

The JCC equates the real marginal cost of hiring a worker, the left hand side, with the

marginal bene�t, the right hand side. Importantly, the marginal bene�t depends positively

on the ratio 1�Jt(with J equal either to P or to Q), which is a positive function of the number

of �rms in the market, Nt. Stronger competition leads to a lower mark up which stimulates

demand by consumers and hence has a positive e¤ect on output and ultimately on employment.

As shown by Colciago and Rossi (2011), a positive technology shock leads to entry of new

�rms and thus to an increase in 1�Jt. In equilibrium, since hiring depends on the current and

expected future values of the marginal product of labor, this boosts hiring and employment

with respect to a model with constant markups.

The JCC is common across �rms, independently of their period of entry. Thus, the optimal

hiring policy of new producers, i.e. �rms which at time t are producing for the �rst time and

have no initial workforce, consists in posting as many vacancies as required to reach the size of

�rms which started production in earlier periods. This has two implications. The �rst one is

that the size-gap between new producers and incumbent �rms is closed in a single period. The

second one is that new producers grow faster than more mature �rms. This is consistent with

the U.S. empirical evidence discussed in Haltiwanger et al. (2010), which suggests that a start-

up creates on average more new jobs than an incumbent �rm. Given vacancy posting is costly,

new producers will su¤er lower pro�ts and pay lower dividends in their �rst period of activity

with respect to �rms which entered into the market in earlier periods. This is consistent with

the evidence on the �nancial behavior of �rms discussed by Cooley and Quadrini (2001).

3.5 Bargaining over Wages and Hours

In the Appendix it is shown that Nash wage bargaining results in the following wage equation

wt = (1� �)b

ht+ �

1

�JtAt + (1� �)�Ct

h1='t

1 + 1='+

��

(1� �)1

htEt�t;t+1�t+1; (18)

where �Jt is the markup function, �t =vtottut

is the tightness of the job market and the para-

meter � re�ects the relative bargaining power of workers. The wage shares costs and bene�ts

associated to the match. The worker is rewarded for a fraction � of the �rm�s revenues and

savings of hiring costs and compensated for a fraction 1 � � of the disutility he su¤ers from

supplying labor and the foregone unemployment bene�ts. The direct e¤ect of competition on

the real wage is captured through the term � 1

�jtAt, which represents the share of the marginal

revenue product (MRP) which goes to workers. As discussed above, entry leads to an increase

in the ratio 1

�jtand hence in the MRP. Thus, everything else equal, stronger competition shifts

the wage curve up. This result is similar to that in Blanchard and Giavazzi (2003), who �nd

a positive e¤ect of competition on the real wage. Hours are set to maximize the joint surplus

of the match. This is obtained when the marginal rate of substitution between hours and

9

consumption equals the MRP of labor, that is

�Cth1='t =

1

�JtAt: (19)

Stronger competition leads to an increase in hours bargained between workers and �rms for

the same reasons for which competition positively a¤ects the wage schedule.

3.6 Aggregation and Market Clearing

Considering that the individual workforce, nt, is identical across producers leads to

Lt = ntNt (20)

To obtain aggregate output notice that PtYt =NtXi=1

ptyt = Ntptyt, further givenptPt= 1 and

the individual production function it follows that

Yt = Ntyt = AtLtht = AtHt (21)

where Ht is the amount of total hours worked. As a consequence At amounts to average

labor productivity, which is assumed to follow a �rst order autoregressive process given by

ln (At=A) = �A ln (At�1=A)+ "At, where �A 2 (0; 1) and "At is a white noise disturbance, withzero expected value and standard deviation �A.

Aggregating the budget constraints of households we obtain the aggregate resource con-

straint of the economy

Ct + Net =WthtLt +�t (22)

which states that the sum of consumption and investment in new entrants must equal the sum

between labor income and aggregate pro�ts, �t, distributed to households at time t. Goods�

market clearing requires

Yt = Ct +NEt + �v

tott (23)

where vtott is the sum of vacancies posted by new entrants and by �rms which entered in earlier

periods. Finally, the dynamics of aggregate employment reads as

Lt = (1� �) %Lt�1 + qtvtott (24)

which shows that workers employed to a �rm which exits the market join the mass of unem-

ployed.

4 Calibration

To solve the model described in the previous section the equations are linearized around the

model�s steady state.8 Calibration is as follows. The discount factor, �, is set to 0.99. As in8The resulting linearized system is solved using DYNARE.

10

BGM (2012) the rate of business destruction, �, equals 0.025. This means roughly 10 percent

of �rms disappear from the market every year, independently of �rm age. The entry cost is

= 1 and held constant along the cycle. With no loss of generality, the value of � is such

that steady state labor supply equals one. The Frisch elasticity of labor supply is ' = 1. The

intersectoral elasticity of substitution is " = 6, as estimated by Christiano, Eichenbaum and

Evans (2005). As standard in the literature we set the steady state marginal productivity

of labor, A, to 1. We calibrate the parameters of the productivity process as estimated by

Rios-Rull and Santaeulàlia-Llopis (2010), with persistence �A = 0:958 and standard deviation

�A = 0:0067. We set the separation rate % equal to 0:1, as suggested by estimates provided

by Hall (1995) and Davis et al. (1996). The elasticity of matches to unemployment, ; is set

equal to the worker bargaining power � and is equal to 12 ; as in the bulk of the literature. The

e¢ ciency parameter in matching, m, and the steady state job market tightness are calibrated

to target an average job �nding rate, z, equal to 0.7 and a vacancy �lling rate, q, equal to

0.9. We draw the latter value from Andolfatto (1996) and Den Haan et al. (2000), while

the former from Blanchard and Galì (2010).9 Finally, we calibrate the unemployment bene�t

in real terms, b, such that the monetary replacement rate, bwh , equals 0:60. This value is

consistent with that reported in the OECD Economic Outlook of 1996 for the US. Given these

parameters we can recover the cost of posting a vacancy � by equating the steady state version

of the JCC and the steady state wage setting equation. Notice that none of the qualitative

result is a¤ected by the calibration strategy.

5 Productivity Shocks and Dynamics of the Labor share

In what follows we study the impulse response functions of the labor share and its components

to a one standard deviation increase in technology.10 To isolate the role of endogenous markup

variability for the dynamics of the labor share we compare the performance of the models

with Bertrand and Cournot competition to that of a model characterized by monopolistic

competition. Under monopolistic competition �rms do not interact strategically and set a

constant markup over marginal costs equal to � = ""�1 .

Figure 2 shows that, on impact, the real wage increase less than average labor productivity

no matter the form of competition in the goods market. As argued above, Nash bargaining

delivers the countercyclicality of the labor share of income. Under monopolistic competition,

after peaking on impact, the real wage returns monotonically to its initial level. Further, it

never rises relative to labor productivity. As a result the labor share does not overshoot.

9A job �nding rate equal to 0.7 corresponds, approximately, to a monthly rate of 0.3, consistent with US

evidence.10This is for consistency with the evidence displayed in Figure 1.

11

0 10 20 30 400

0.2

0.4

0.6

Cou

rnot

 Com

petiti

on

0 10 20 30 40

­0.1

0

0.1

0.2

0 10 20 30 400

0.2

0.4

0.6

Bertr

and 

Com

petiti

on

0 10 20 30 40

­0.1

0

0.1

0.2

0 10 20 30 400

0.2

0.4

0.6

Mon

opol

istic 

Com

petiti

on

0 10 20 30 40

­0.1

0

0.1

0.2

Labor productivity Real wage Labor share Price markup

Figure 2: Impulse response functions to a technology shock. Top panel: Cournot competition;

middle panel: Bertrand competition; bottom panel: monopolistic competition.

This is not the case when the goods market is characterized by oligopolistic competition.

Under both Bertrand and Cournot, the labor share is countercyclical due to Nash Bargaining.

Moreover, the labor share overshoots its long run level after about �ve quarters, it peaks at

about the �fth year at a level larger than its long-run value and seven years after the shock

has hit the economy is still halfway toward its average. The key lies in the countercyclical and

inertial response of the price markup. To see this, consider the log-deviations of the real wage

and labor hours from their steady state. These are respectively

wt = �1

�At � �t

���2ht +�3Et�t+1 (25)

and

ht = '�At � �t � ct

�; (26)

where �1 = 1�w

��+'1+'

�, �2 = 1��1, �3 = ���

w and �t+1 = b�t;t+1 + b�t+1. Under all plausibleparametrization, we �nd that �1 is lower than one. As a result, only a fraction �1 < 1 of the

impact increase in productivity At goes to workers. Further, equation (26) shows that labor

hours increase with productivity and contribute to dampen the positive e¤ect of productivity

on real wages. Hence, the impact increase in real wages is lower than that of labor productivity

and the labor share is countercyclical. In a model with endogenous market structures these

are just partial e¤ects. Technology shocks create expectations of future pro�ts which lead to

the entry of new �rms. Stronger competition leads to lower price markups. Given that entry

is subject to a one period time-to-build lag, the total number of �rms, Nt, does not change on

impact, but builds up gradually. As shown in Figure 2, in the Cournot and in the Bertrand

model this translates into an initially muted response of the markup. As entry increases the

12

number of �rms, however, the price markup starts declining. In particular it �nds its negative

peak after few periods and then gradually reverts to its long run value.11 Equation (25)

shows that a persistently lower markup acts as a shifter of the standard marginal product of

labor allowing the real wage to rise relative to the average productivity of labor for several

periods. Since blst = wt � At; this explains the overshooting of the labor share. Thus, we can

state that the dynamic response of the markup to technology shocks is fundamental for the

overshooting.12

In the Cournot model the initial drop of the labor share as well as the timing and amplitude

of the overshooting are very close to their data counterpart (see Figure 1). The response of the

real wage is also quantitatively and qualitatively similar to the empirical one. Di¤erently, in

the Bertrand model the magnitude of the overshooting is lower than in the data. The reason

is the stronger markup variation under Cournot, which is re�ected in a larger wedge between

the real wage and average labor productivity.

5.1 The role of real wage rigidity

Aggregate wages are characterized by an high degree of persistence, so that sudden and large

shifts in the aggregate wage level are not observed. The existence of real wage rigidities has

been pointed to by many authors as a feature needed to account for a number of labor market

facts (see, e.g., Hall 2005).

Real wage rigidity leads to a slow adjustment of wages to labor market conditions. In par-

ticular, in response to a productivity shock it leads to a smoother and more inertial dynamics

of the real wage than the average labor productivity. As emphasized above, this is the key

feature a model should satisfy to address the overshooting of the labor share in response to a

technology shock. For this reason we study the e¤ect of real wage rigidity on the dynamics of

the labor share. Following Hall (2005), we model real wage rigidity in the form of a backward

looking social norm:13

wt = �wwt�1 + (1� �w)wnasht (27)

where �w is an index re�ecting the degree of real wage rigidity and wnasht is the wage obtained

under Nash Bargaining, i.e. that in equation (18). Notice that �w = 1 implies a �xed real wage,

while �w = 0 corresponds to the case of Nash bargaining analyzed earlier. As observed by

Blanchard and Galì (2007), equation (27), even though admittedly ad-hoc, is a parsimonious

11Notice that the shape of the response of the price markup to a technology shock is consistent with the

evidence in Rotember and Woodford (1999) and the VAR evidence in Colciago and Etro (2010).12We consider alternative values of � and ' and we �nd that they do not alter qualitatively the overshooting

result. This holds also in the case with �xed individual hours, that is with ' = 0:13Blanchard and Galì (2007), Christo¤el and Linzert (2010), Ascari and Rossi (2011) and Faia and Rossi

(2012) take a similar approach.

13

0 10 20 30 40­0.8

­0.6

­0.4

­0.2

0

0.2

Cournot Competit ion

0 10 20 30 40­0.8

­0.6

­0.4

­0.2

0

0.2

Bertrand Competition

0 10 20 30 40­0.8

­0.6

­0.4

­0.2

0

0.2

Monopolistic Competition

φw=0.5 φw=0.9

Figure 3: Labor share response to a technology shock under alternative degrees of real wage

rigidity. Left panel: Cournot competition; middle panel: Bertrand competition; right panel:

monopolistic competition.

way of introducing a slow adjustment of real wages to labor market conditions.14

Figure 3 displays the response of the labor share to a one standard deviation increase in

technology in the Bertrand and the Cournot models as well as in the model with monopolistic

competition. Since there is no evidence on the degree of real wage rigidities, we consider two

alternative values of the parameter �w. Dashed lines refer to the case �w = 0:5, the midpoint

of the admissible range. Solid lines depict the extreme case where �w = 0:9.15

In the model with constant price markups the labor share overshoots its long run level

just in the case of extreme real wage rigidity. Nevertheless the overshooting is negligible. This

con�rms that countercyclical price markups are key for the overshooting of the labor share.

Augmenting the Cournot and Bertrand competitive frameworks with a limited degree of

real wage rigidity, does not alter the previous �ndings substantially, nevertheless it improves

the matching of the amplitude of the overshooting from a quantitative point of view. Our

view is that real wage rigidity does not seem to play a crucial role for the dynamics of the

labor share of income.14The authors consider alternative formalizations, explicitly derived from staggering of real wage decisions.

Although the algebra is more involved, the basic conclusions are the same as those obtained with the ad-hoc

formulation.

15A value of �w = 0:9 implies a real wage adjustment of about 6 quarters.

14

6 Conclusion

Recent U.S. evidence suggests that the response of labor share to a productivity shock is

characterized by countercyclicality and overshooting. To account for these empirical �ndings,

a model should satisfy two desiderata. The �rst one is that the impact increase in the real

wage must be lower than that of average labor productivity. The second one is the presence

of a persistent wedge between average labor productivity and real wages such that, in the

aftermath of the shock, the response of the latter raises above that of the former for several

periods.

We propose a DMP model characterized by �rms�entry and oligopolistic competition be-

tween producers that addresses this evidence. Nash bargaining delivers the countercyclicality

of the labor share in response to a technology shock. The countercyclicality of price markup

originating from strategic interactions in the goods market acts as a shifter of the standard

marginal product of labor and allows the labor share of income to overshoot.

While real wage rigidity helps accounting for a number of labor market facts, such as the

variability of unemployment in response to a technology shock and the slow response of real

wages to labor market conditions, it does not seem to play a crucial role for the dynamics of

the labor share of income.

Appendix

Let us provide some terminology before starting the analysis. The term new entrants refers to the �rms

which enter the market at time t. The value of these �rms is denoted by V et . The term new producers

refers to �rms which entered the market in t-1 and at time t produce for the �rst time (these �rms are

a fraction (1� �) of time t-1 new entrants). The term incumbent �rms refer to �rms which entered

the market in period t-2 or earlier. Notice that new producers and incumbent �rms have the same

value, which we denote with Vt. This is so since new producers close their size gap with incumbent

�rms in their �rst period of activity. For this reason after their �rst period of activity new producers

are indistinguishable from �rms that entered in t-2 or earlier.

Households

We assume that households invest in both incumbent �rms and new entrants. Bonds and stocks are

denominated in terms of the �nal good. The budget constraint expressed in nominal terms is

PtBt+1+P tCt+P t

Z 1

0VjtNjtsjt+1dj + P t

Z 1

0V ejtN

ejts

ejt+1dj

= WtLtht+(1� Lt)Ptb+ (1 + rt)P tBt+(1� �)PtZ 1

0[�jt(";Njt) + Vjt]Njt�1sjtdj+

+(1� �)PtZ 1

0

��newjt (";Njt) + Vjt

�N ejt�1s

ejtdj � P tTt (28)

15

where Bt is net bond holdings with interest rate rt, Vjt is the value of an incumbent �rm in sector

j and V ejt is the value of a new entrant in the same sector. The variables Njt and N ejt represent

the number of active �rms in sector j and the new entrants in this sector at the end of the period,

respectively. The variable sjt represents the share of the portfolio of incumbent �rms belonging to

sector j that is owned by the household, while sejt is the share of portfolio of new entrants held by the

household. The term (1� �)PtR 10 [�jt(";Njt) + Vjt]Njt�1sjt represents the sum between the value

of the portfolio of �rms which entered the market in period t-2 or earlier held by the household and the

pro�ts distributed by these �rms. Notice the number of these �rms is equal to (1� �)Njt�1 in eachsector. The term (1� �)Pt

R 10

h�newjt (";Njt) + Vjt

iN ejt�1s

ejt denotes the sum between the value of

the portfolio of new producers, where (1� �)N ejt�1 is the number of �rms which produce for the �rst

time at time t. In the budget constraint we have imposed the symmetry in the value of new �rms and

incumbent �rms. Finally PtTt represent nominal lump sum taxes imposed to �nance unemployment

bene�ts. The household recognizes that employment is determined by the �ows of its members into

and out of employment according to

Lt=(1� �) %Lt�1+ztut (29)

Equations (28) and (29) represent the constraint to the utility maximization problem. We denote with

�t the Lagrangian multiplier of the �rst constraint, while �t is the one of the second constraint.

The intertemporal optimality conditions with respect to sjt+1, sejt+1 for each sector, and with

respect to Bt+1 are, respectively

PtVjt= �Et (1� �)�t+1�t

Pt+1 [�jt+1(";Njt+1) + Vjt+1] (30)

PtVejt= �Et (1� �)

�t+1�t

Pt+1��newjt+1(";Njt+1) + Vjt+1

�(31)

Pt�t= �Et(1 + rt+1)P t+1�t+1 (32)

The optimal choice of consumption requires

1

PtCt= �t (33)

Notice that �t has the meaning of the marginal value to the household of having a member employed

rather than unemployed. The latter a¤ects bargaining over the real wage and individual hours and it

is given by

�t=1

Ct(wtht � b)��

h1+1='t

1 + 1='+�Et [(1� �) �� zt+1] �t+1 (34)

where wt =WtPtis the real wage.

16

Pro�t Maximization Problem

Consider Bertrand competition. We initially consider the problem of an incumbent �rm. Substituting

the direct demand for the individual good into period t real pro�ts, we obtain

�t=pt(i)

1�""NtXi=1

pt(i)�("�1)

#EXP tPt

�wt (i)nt (i)ht (i)��vt (i) (35)

The pro�t maximization problem of an incumbent �rm reads as

maxfpt(i);nt(i);vt(i)g1t

�t+Et

1Xs=t+1

�t;s�s (36)

subject to

Atnt (i)ht(i) =pt(i)

�"EXP t"NtXi=1

pt(i)(1�")

# (37)

and

nt (i)= �nt�1 (i)+vt (i) qt (38)

Lagrangian multipliers on constraints (37), and (38) are respectively mct (i) and �t (i). Setting up

the Lagrangian L, the FOCs with respect to nt (i), vt (i) and pt (i) are, respectively

@L@nt (i)

= 0 : wt (i)ht (i)+�t (i)�mct (i)Atht (i)= %Et�t;t+1�t+1 (i) (39)

@L@vt (i)

= 0 : � = �t (i) qt (40)

and

@L@pt (i)

= 0 :

(1� ")"NtXi=1

pt(i)(1�")

#� (1� ") pt(i)1�""

NtXi=1

pt(i)1�"

#2 pt(i)�"EXP t

Pt+

mct (i)

"pt(i)�1

"NtXi=1

pt(i)(1�")

#+ (1� ") pt(i)�""

NtXi=1

pt(i)1�"

#2 pt(i)�"EXP t

= 0 (41)

Notice that we assume that �rms take individual wages as given when choosing employment. Also

notice that since there is a continuum of sectors, the individual �rm takes the aggregate price level

as given. The second condition shows that �t (i), the surplus created by a match, is identical across

incumbent �rms. Before providing an explicit formula for the individual price level and the price

17

markup, we turn to the pro�t maximization problem of a �rst period producer which sets the price

for the �rst time. The relevant di¤erence with respect to the previous case is represented by the form

of constraint (38) which reads as vt (i) qt = nt (i), since producers in their �rst period of activity

have no initial workforce. However, FOCs with respect to pt(i), nt (i) and vt (i) are identical to those

reported above. Since the surplus �t created by a match is identical across all producers , they will

face the same wage bargaining problem, thus will face the same wage, wt (i) = wt, the same marginal

cost, mct (i) = mct, and will demand the same amount of hours, ht (i) = ht. As a result the third

condition can be written as

(1� ")NtP 1�"t � (1� ") pt (i)1�"=MCt

h("� 1) pt (i)�" � "pt (i)�1NtP 1�"t

i(42)

where MCt (= Ptmct) is the nominal marginal cost, which shows that pt (i) does not depend on any

�rm speci�c variable. In other words all �rms which are active at time t, no matter the period of entry,

choose the same price. Since �rms face the same demand function and adopt the same technology, it

follows that yt (i) = yt and nt (i) = nt: We are now ready to provide an expression for the common

price chosen by �rms. Given that �rms choose the same price level, it follows that p (i) = pt = Pt.

Imposing symmetry and rearranging, condition (14) can be rewritten as

1

mct= �t (43)

where

�t=" (Nt � 1) + 1("� 1) (Nt � 1)

(44)

Further, notice that, after imposing symmetry, by combining equation (39) and (40) we get the JCC

reported in the main text. Under Cournot competition pro�t maximization must take the inverse

demand function as a constraint. The latter is

pt(i) =yt(i)

� 1"

NtXj=1

yt(j)"�1"

EXPt

which implies that period pro�ts can be written as

�t=yt(i)

1� 1"

NtXj=1

yt(j)"�1"

EXPtPt

�wt (i)nt (i)ht (i)�kvt (i)

Setting up a Lagrangian function as in the previous case and di¤erencing with respect to yt(i); nt (i) ; vt (i),

it can be easily veri�ed that the FOCs with respect to nt (i) ; vt (i) are unchanged with respect to the

Bertrand case.

18

Wage setting

The real wage and hours worked are set to maximize the product

(�t)1�� (�tCt)

� (45)

where the term in the �rst bracket, �t; is the value to the �rm of having an additional worker, i.e.,

�t=1

�tAtht�wtht+%Et�t;t+1�t+1 (46)

the second term, �t; is the household�s surplus expressed in units of consumption,

�t=1

Ctwtht��

h1+1='t

1 + 1='� b

Ct+�Et [(1� �) �� zt+1] �t+1 (47)

The FOC with respect to the wage is

(1� �) (�t)�� (�tCt)�d�

dw+� (�tCt)

��1 (�t)1�� d�t

dwCt= 0 (48)

Notice that d�tdwtCt = � d�t

dwt= ht, thus (48) can be simpli�ed as follows

��t=(1� �) �tCt (49)

Multiplying both sides of equation (49) by %� (1� �) Ct�1Ctyields

�%� (1� �) Ct�1Ct

�t=(1� �) %� (1� �)Ct�1�t; (50)

leading one period and taking expectations as of time t leads to

�%Et�t;t+1�t+1=(1� �) %� (1� �)CtEt�t+1; (51)

substituting for �t and �tCt and simplifying

�1

�tAtht= wtht� (1� �)

�h1+1='t Ct1 + 1='

+ b+ �Etzt+1�t+1Ct

!: (52)

Multiplying both sides of (49) by ztCt�1Ct

, leading one period and taking expectation as of time t, we

can rewrite

�zt+1CtCt+1

�t+1=(1� �) zt+1Ct�t+1; (53)

using the latter it follows that

(1� �)�CtEtzt+1�t+1= ��EtCtCt+1

zt+1�t+1=�

(1� �)�t;t+1zt+1�t+1; (54)

substituting into (52) delivers

�1

�tAtht= wtht� (1� �)�

h1+1='t Ct1 + 1='

+(1� �) b+ �

(1� �)�t;t+1zt+1�t+1: (55)

19

Finally, using �t =�qtand zt

qt= �t; and rearranging, we get

wtht=(1� �) b+ �At1

�tht+(1� �)�

h1+1='t

1 + 1='Ct+

��

(1� �)Et�t;t+1�t+1; (56)

which is the wage equation in the text. Similarly, the FOC for hours Nash bargaining is

(1� �) (�t)�� (�tCt)�d�

dh+� (�tCt)

��1 (�t)1�� d�t

dhCt= 0: (57)

Considering that d�tdht= 1

�tAt�wt; and that d�tdht

Ct = wt��h1='t Ct, equation (57) can be written as

(1� �) �tCt�1

�tAt � wt

�+��t

�wt � �h1='t Ct

�= 0: (58)

Finally, using equation (49), equation (58) simpli�es to

ht=

�1

�

�t�t

AtCt

�'(59)

which is the equation for hours worked in the text.

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